Correct, you will not get a United Arrivals invitation. I don't know if BMI has facilities, but Heathrow airport has public showers available. United flies into termnal 3 at Heathrow and according to the Heathrow website http://www.baa.co.uk/main/airports/heathrow/ there are also showers in Terminal 1 (where bmi is) and in the flight connections center.



Showers
Showers are available in the following locations throughout the airport:
Terminal 1
Before security

Terminal 3
Departure lounge - these facilities are currently unavailable due to development works.

Terminal 4
Available in the toilet areas on the Arrivals level, before security and further facilities are available after security control

Additional shower facilities are available in the Flight Connections Centre.